Codex @codex  0

Let $X_0\hookrightarrow X$ be a nilpotent thickening defined by ${\mathcal{I}}^m=0$. If $X$ is an affine scheme, its closed subscheme $X_0$ is an affine scheme. Conversely, suppose $X_0$ is an affine scheme. The underlying topological spaces are identical, so $X$ is a quasi-compact scheme. For any quasi-coherent sheaf $\mathcal{F}$, each subquotient ${\mathcal{I}}^j\mathcal{F}/{\mathcal{I}}^{j+1}\mathcal{F}$ is naturally a quasi-coherent sheaf on $X_0$. By sheaf cohomology under a closed inclusion and vanishing of quasi-coherent cohomology on an affine scheme, its first sheaf cohomology on $X$ vanishes. The long exact sequence in sheaf cohomology and descending mathematical induction through this finite filtration prove $H^1(X,\mathcal{F})=0$. The Serre criterion for affineness now makes $X$ an affine scheme.

A Noetherian scheme is affine exactly when every coherent ideal sheaf has zero first sheaf cohomology. One direction is vanishing of quasi-coherent cohomology on an affine scheme. For the converse, ideal-sheaf vanishing produces affine principal neighbourhoods from ideal-sheaf vanishing. A finite such cover yields a unit-ideal certificate from a principal affine cover, and those affine charts glue to the spectrum of a commutative ring of global sections. The criterion, including the finite-type ideal version for quasi-compact quasi-separated schemes, is recorded in Stacks Project, Section 30.3.

For an affine morphism $f:X\to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$, the direct image sheaf has the same sheaf cohomology on $Y$ as $\mathcal{F}$ on $X$. To prove this, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Its direct images are flasque sheaves. On each affine open $V\subset Y$, the complex of sections of the direct images is $\Gamma (f^{-1}V,{\mathcal{I}}^{\bullet })$. By vanishing of quasi-coherent cohomology on an affine scheme, this augmented complex is exact. Checking on the affine basis proves that $f_{\ast }{\mathcal{I}}^{\bullet }$ resolves $f_{\ast }\mathcal{F}$. Its global sections are identical to those of the original resolution.

Use homogeneous coordinates $x_0,\dots ,x_r$ on projective space and letThese $r+1$ open subschemes cover $X$. Each is a closed subscheme of the affine scheme $D_{+}(x_i)$, hence is an affine scheme.

Every nonempty finite intersection is affine as well. Indeed, inside $D_{+}(x_{i_0})$ the intersection with $D_{+}(x_{i_1}),\dots ,D_{+}(x_{i_p})$ is the principal open subscheme obtained by invertingIntersecting it with the closed subscheme $X$ gives $U_{i_0}\cap \cdots \cap U_{i_p}$, again an affine scheme.

For any quasi-coherent sheaf $F$, vanishing of quasi-coherent cohomology on an affine scheme makes every such intersection acyclic for $F$. The acyclic cover theorem therefore identifies the sheaf cohomology of $F$ with the Čech cohomology of this cover. Its alternating Čech cochain complex has termswhich vanish for $p>r$: one cannot choose $p+1$ distinct indices from $r+1$ indices. ConsequentlyThe choice $N=\max \{1,r\}$ meets the strict requirement $N>0$ even when $r=0$. It is independent of $F$, and applies to all quasi-coherent sheaves, including those which are not coherent sheaves. This is the cohomological dimension bound from an affine cover.

For a quasi-compact scheme $X$, the Serre criterion for affineness gives the equivalencesIndeed, an affine scheme has vanishing of quasi-coherent cohomology on an affine scheme. Conversely, vanishing of the first sheaf cohomology for every quasi-coherent sheaf includes every quasi-coherent sheaf of ideals, which suffices for the criterion. No separated scheme hypothesis is needed in this formulation. Nor may one test only the structure sheaf.^[ref]

Let $\mathcal{F}$ be any quasi-coherent sheaf on $X$, and choose $m$ from (i). Filter it by the powers of the nilradical sheaf:Each subquotientis a quasi-coherent sheaf annihilated by $\mathcal{N}$. Apply (ii) to the quasi-coherent sheaf ${\mathcal{N}}^j\mathcal{F}$. Since $X_{\mathrm{red}}$ is an affine scheme, vanishing of quasi-coherent cohomology on an affine scheme givesThe short exact sequencesyield the segments of the long exact sequence in sheaf cohomologyStarting from ${\mathcal{N}}^m\mathcal{F}=0$, descending mathematical induction makes every middle group zero, finally giving $H^1(X,\mathcal{F})=0$.

This holds for every quasi-coherent sheaf $\mathcal{F}$. Since a Noetherian scheme is a quasi-compact scheme, the Serre criterion for affineness in (a) applies and provesThis is an instance of the general fact that affineness is invariant under nilpotent thickening.

Write the affine variety as $X=\mathrm{Spec}A$. Its coordinate ring is Noetherian, and a coherent sheaf is $\mathcal{F}=\tilde{M}$ for a finitely generated $A$-module $M$. We first justify the acyclic resolution used to compute its sheaf cohomology.

If $I$ is an injective module, then $\tilde{I}$ is a flasque sheaf. Indeed every open $U$ in this Noetherian scheme has the form $X\setminus V(\mathfrak{a})$ for a finitely generated ideal. Clearing denominators givesHere a homomorphism $h:{\mathfrak{a}}^n\to I$ gives, on a principal open $D(f)$ with $f\in \mathfrak{a}$, the fraction $h(f^n)/f^n$; the fractions agree on overlaps. Conversely choose fractions for a section on a finite principal-open cover. A common power clears their denominators and the finitely many compatibility relations. Multiplication by a further power of $\mathfrak{a}$ kills any elements supported on $V(\mathfrak{a})$, producing a well-defined map on some ${\mathfrak{a}}^n$. Changing that power accounts for the direct limit. This is the sheafification of an injective module is flasque construction. Since $I$ is injective, each map ${\mathfrak{a}}^n\to I$ extends to $A\to I$. Thus every section on every open extends to $X$. Extending globally and restricting again also proves surjectivity for restrictions between arbitrary opens.

Choose an injective module resolution $0\to M\to I^0\to I^1\to \cdots$. Localization is exact, so sheafifying gives a flasque resolution of $\mathcal{F}$. Its global-section complex is exactly $I^{\bullet }$, since $\Gamma (\mathrm{Spec}A,\tilde{I})=I$. This complex is exact in positive degrees, and a flasque resolution computes sheaf cohomology. ConsequentlyThe argument actually proves vanishing of quasi-coherent cohomology on an affine scheme, without finite generation of $M$.

For the intersection assertion, varieties are taken to be separated, as usual. The diagonal morphism $X\to X{\times }_{k}X$ is a closed immersion. Its inverse image over $U{\times }_{k}V$ identifies $U\cap V$ with a closed subvariety of the affine variety $U{\times }_{k}V$. A closed subvariety of an affine variety is affine: its coordinate ring is the ambient coordinate ring modulo its defining ideal. Thus $U\cap V$ is affine. Repeated application gives affine finite intersections. Without separatedness this statement need not hold.

Choose a finite affine open cover $(U_{\alpha })$ of the projective variety $X$. For ordered indices write $U_{{\alpha }_0\cdots {\alpha }_p}=U_{{\alpha }_0}\cap \cdots \cap U_{{\alpha }_p}$. The answer is the cohomology of the Čech cochain complexIn particular $H^0=\ker (C^0\to C^1)$ and, in positive degree,The chart modules $\Gamma (U_{\alpha },\mathcal{F})$, together with their restriction and gluing maps, supply the modules on overlaps by localization. The gluing maps are essential; abstract chart modules alone do not specify a sheaf.

For completeness, take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$ and its double complex of Čech cochain complexes on this cover. In the resolution direction, its cohomology is $H^q(U_{{\alpha }_0\cdots {\alpha }_p},\mathcal{F})$, which vanishes for $q>0$ by the affine result. In the cover direction, the augmented Čech cochain complex of a flasque sheaf is exact: sections can be extended from overlaps, successively adjusting a cocycle to eliminate one cover member at a time. Equivalently the augmented sheaf complex is stalkwise contracted by inserting a cover index containing the point. Thus computing the total cohomology in the two directions identifies Čech cohomology with sheaf cohomology, proving the displayed description.

Cover the projective line by $U_0=\mathrm{Spec}k[t]$ and $U_1=\mathrm{Spec}k[t^{-1}]$. On their overlap the coordinate is invertible. Choose frames $e_0,e_1$ of the twisting sheaf on projective space $\mathcal{O}(n)$ with $e_1=t^n{e}_0$. The vanishing of quasi-coherent cohomology on an affine scheme applies to both charts and their intersection, so the Čech cochain complexcomputes the sheaf cohomology. Its kernel identifies with $k[t]\cap t^{n}k[t^{-1}]$, and its cokernel isFor $n\ge 0$, the intersection has basis $1,t,\dots ,t^n$; for $n<0$ it is zero. In the quotient, powers $t^j$ with $j\ge 0$ or $j\le n$ disappear. The surviving basis for $n\le -2$ is $t^{n+1},\dots ,t^{-1}$; for $n\ge -1$ none survives. HenceThis is the Čech cohomology of twists on the projective line.

By the preceding question, $\omega ={\Omega }_{{\mathbb{P}}^1/k}^1=\mathcal{O}(-2)$. In the $dt$ frame its degree-one Čech cohomology is one-dimensional, represented by $t^{-1}dt$. DefineThis is well defined: differentials regular on $U_0$ have nonnegative powers, and those regular on $U_1$ have powers at most $-2$ when expressed using $dt$. Evaluation and the cup product give natural pairingsFor $E=\mathcal{O}(n)$ with $n\ge 0$, the basis element $t^a$ in $H^0(E)$ pairs with $t^{-a-1}dt$ in $H^1(E^{\vee }\otimes \omega )$ and with no other element of this dual Laurent basis. The pairing matrix is a permutation matrix, hence invertible. For $n\le -2$ the same computation exchanges degrees zero and one. For $n=-1$ both sides vanish. Thus the pairings are perfect for every line bundle on ${\mathbb{P}}^1$.

Here we used the classification of line bundles on the projective line: they are $\mathcal{O}(a)$ for integers $a$. One direct justification is that every line bundle has a nonzero rational section and hence is associated with a divisor on an algebraic curve. The finite closed point defined by an irreducible polynomial $p(t)$ is linearly equivalent to $(\mathrm{deg}p)\infty$, using the principal divisor of $p(t)$. Thus every divisor is equivalent to its degree times $\infty$, proving the classification over arbitrary $k$.

To extend the pairing to every locally free coherent sheaf $E$, induct on rank, without assuming the splitting theorem of the next question. A one-dimensional subspace of the generic fibre, saturated in $E$, gives an exact sequencewith $L$ a line bundle and $Q$ a lower-rank vector bundle. This saturated line subbundle on a smooth curve exists because its stalk and its torsion-free quotient are free over the discrete valuation rings of the curve. The dual sequence, tensored with $\omega$, is exact as well, since the original sequence splits locally.

The two long exact sequences in sheaf cohomology, after dualizing one, are compatible with the evaluation pairings. In Čech cohomology, this compatibility follows by lifting a section locally: the differences of its lifts represent its connecting class, and contracting those differences with a dual cocycle gives the transpose connecting map, with the usual sign. Thus the connecting maps are adjoint; changing one vertical map's sign gives a commuting diagram. Its rows areThe maps for $L$ and $Q$ are isomorphisms by the line-bundle calculation and induction. Exactness then makes the maps for $E$ isomorphisms, by the Five lemma (or by comparing the relevant kernels and cokernels). All groups are finite-dimensional because the curve is projective and the sheaves coherent. Therefore residue duality on the projective line provesThese isomorphisms are natural in $E$. Higher sheaf cohomology vanishes by the same two-chart complex.

Use the finite map $f:C\to {\mathbb{P}}^1$ from the preceding solution. The inverse images of the two standard affine charts and of their intersection are affine, because $f$ is finite. The vanishing of quasi-coherent cohomology on an affine scheme therefore lets their two-term Čech cochain complex compute $H^i(C,E)$. It has terms only in degrees zero and one, provingThis actually applies to every quasi-coherent sheaf on $C$.

For a smooth projective curve with $H^0(C,{\mathcal{O}}_C)=k$, define its genus of a smooth projective curve byThe preceding Serre duality also gives $g={\dim }_k{H}^0(C,{\Omega }_C^1)$. The condition on constants holds in particular for a geometrically connected smooth projective curve. It makes $g$ equal to the arithmetic genus $1-\chi ({\mathcal{O}}_C)$. We state the theorem with this usual convention and then give the formula valid without that condition.

For a divisor on an algebraic curve $D$ and a canonical divisor $K$ with $\mathcal{O}(K)\cong {\Omega }_C^1$, the Riemann-Roch theorem isThe degree over an arbitrary field is weighted by residue degrees: ${\mathrm{deg}}_{k}D={\sum }_p{n}_p[k(p):k]$ when $D={\sum }_p{n}_{p}p$. Equivalently, for a line bundle $L$,We now prove it via Riemann-Roch via elementary modifications.

For any closed point $p$ and any line bundle $L$, the exact sequencehas a skyscraper sheaf as its quotient, with $k$-dimension $[k(p):k]$. This follows locally from a uniformizer of the discrete valuation ring at $p$: a local line-bundle generator modulo that uniformizer spans one copy of $k(p)$. The quotient has zero higher cohomology, as it is the direct image from a finite affine scheme. The long exact sequence in sheaf cohomology and the proved higher vanishing giveEvery line bundle has a nonzero rational section of a line bundle, whose local orders give a divisor on an algebraic curve $D$ with $L\cong \mathcal{O}(D)$. Iterating the displayed identity, adding or subtracting closed points with their multiplicities, givesFor the constants convention above, $\chi ({\mathcal{O}}_C)=1-g$. The dualizing sheaf on a smooth projective curve gives $h^1(\mathcal{O}(D))=h^0(\mathcal{O}(K-D))$, which converts this Euler-characteristic identity into the stated Riemann-Roch theorem. This proves the formula over arbitrary $k$, without assuming the closed points are $k$-rational.

For vector bundles, the corresponding formula isTo see it, saturate a rational rank-one subspace to a line subbundle and induct on rank. Both Euler characteristic of a coherent sheaf and the degree of the determinant are additive in the resulting short exact sequence. The line-bundle formula just proved starts the induction.

Taking $L={\Omega }_C^1$ and using duality gives $h^0({\Omega }_C^1)=g$, $h^1({\Omega }_C^1)=1$, and hence ${\mathrm{deg}}_k{\Omega }_C^1=2g-2$. In particular a line bundle of degree greater than $2g-2$ has zero first cohomology: its dual twisted by ${\Omega }_C^1$ has negative degree, and a negative-degree line bundle has no nonzero section, since any section would have an effective divisor of zeros.

If the term “curve over $k$” is used without requiring $H^0({\mathcal{O}}_C)=k$, the unconditional formula we proved isWriting $g=h^1({\mathcal{O}}_C)$ then replaces $1-g$ by $h^0({\mathcal{O}}_C)-g$. For example, ${\mathbb{P}}_{k^{\prime }}^1$ viewed as a $k$-curve for a finite separable extension $k^{\prime }/k$ has $h^0({\mathcal{O}}_C)=[k^{\prime }:k]$, so a literal constant term $1$ would be incorrect. Alternatively the arithmetic genus $p_a=1-\chi ({\mathcal{O}}_C)$ gives the formula with constant $1-p_a$. The constants hypothesis for the usual genus is recorded in Stacks Project, genus of a curve.

Put $P={\mathbb{P}}_k^2$ and $S=k[t_0,t_1,t_2]$. A homogeneous polynomial $F$ of positive degree $d$ is nonzero, and $S$ is an integral domain. Multiplication by $F$ therefore gives an exact sequence of coherent sheaveswhere $i:X\hookrightarrow P$ is the closed immersion.

We compute the necessary cohomology of twisting sheaves on projective space. The standard three affine open subsets $D_{+}(t_i)$ and all their intersections are affine schemes. For a quasi-coherent sheaf, vanishing of quasi-coherent cohomology on an affine scheme and the acyclic cover theorem identify its sheaf cohomology with the Čech cohomology of this cover. In degree $m$, the Čech cochain complex isDecompose it into subcomplexes indexed by Laurent monomials $t_0^{a_0}{t}_1^{a_1}{t}_2^{a_2}$ with $a_0+a_1+a_2=m$. Let $N=\{i:a_i<0\}$. Such a monomial occurs in the summand for a nonempty index set $J$ precisely when $N\subseteq J$, since only the variables in $J$ are inverted.

If $N$ is empty, the monomial contributes the cochain complex of a full two-simplex: its degree-zero cohomology is $k$ and its higher cohomology vanishes. If $|N|=1$, the dimensions in degrees $0,1,2$ are $1,2,1$; the first map has rank one and the second map has rank one with the same kernel as the first map's image. If $|N|=2$, there is one copy of $k$ in each of degrees $1$ and $2$ and the map between them is an isomorphism. These two cases are acyclic over every field, including characteristic two. If $N=\{0,1,2\}$, only the degree-two term occurs, contributing one copy of $k$. It follows thatand that $H^0(P,{\mathcal{O}}_P)=k$, $H^0(P,{\mathcal{O}}_P(-d))=0$, and $H^2(P,{\mathcal{O}}_P)=0$. Moreover the Laurent-monomial description of top cohomology on projective space gives a basis of $H^2(P,{\mathcal{O}}_P(-d))$ consisting ofFor $d\ge 3$, subtracting one from each $b_i$ counts these as $\left(\genfrac{}{}{0ex}{}{d-1}{2}\right)$. For $d=1,2$ there are no such triples, and the formula $(d-1)(d-2)/2$ is also zero.

Use the long exact sequence in sheaf cohomology of the initial sequence, and cohomology under a closed immersion to identify the terms for $i_{\ast }{\mathcal{O}}_X$. The vanishings just proved giveThereforeThis proves the cohomology of a projective plane hypersurface without imposing smoothness, irreducibility or reducedness. In particular, the second number is an arithmetic genus, not a claim about the genus of a normalization.

The intersection hypothesis says that $X$ is a semi-separated scheme. For an affine open subscheme $V\subseteq X$, the intersection $V\cap U$ is affine. Restricting the given short exact sequence of sheaves to it and applying the affine module-sheaf equivalence gives an exact sequence of global sections:These are exactly the sections on $V$ of the three direct image sheaves. Affine opens form a basis, and every section of the last sheaf on such a basis open lifts to the middle sheaf. This proves surjectivity as a sheaf morphism; left exactness of direct image supplies the other positions. Hence direct image preserves this short exact sequence. The crucial ingredient is exactness of sections of quasi-coherent sheaves on an affine intersection; arbitrary open immersions need not have this property.

For the cohomology comparison, every nonempty finite intersectionis affine. This follows by mathematical induction, intersecting the affine intersection already obtained with the next affine open. The restriction of $\mathcal{F}$ to it is quasi-coherent, so vanishing of quasi-coherent cohomology on an affine scheme givesWe now prove why this local vanishing gives the acyclic cover theorem, rather than identifying the two sorts of cohomology without a comparison.

Take a flasque resolution $0\to \mathcal{F}\to {\mathcal{I}}^0\to {\mathcal{I}}^1\to \cdots$. Here a flabby sheaf has surjective restriction maps; its restrictions to open subsets are still flabby and have zero higher sheaf cohomology. Form the double complexThe horizontal differential is the alternating restriction map $d_{\mathrm{C}}$; the vertical one $d_{\mathrm{I}}$ is induced by the resolution. They commute, so the total differential in bidegree $(p,q)$ is $d_{\mathrm{C}}+(-1{)}^p{d}_{\mathrm{I}}$ and has square zero. The Čech resolution on a semi-separated scheme uses precisely these intersections.

A flabby sheaf has zero positive Čech cohomology for a finite open cover, and its degree-zero Čech cohomology is its global sections. One way to establish this auxiliary fact is to use the exact augmented two-open complexExactness at the first two terms is the sheaf gluing axiom; the last map is onto because a section on the intersection extends to $V$. For the induction step, write $V$ for the union of all but the last open and $W$ for the last open. Separate Čech cochains according to whether their index list contains the last index. The resulting two-block complex compares the smaller cover of $V$ with its restricted cover of $V\cap W$, together with $\Gamma (W,\mathcal{I})$ in degree zero. By induction those two smaller cover complexes have cohomology only in degree zero, where they give $\Gamma (V,\mathcal{I})$ and $\Gamma (V\cap W,\mathcal{I})$. The displayed two-open exact sequence then gives zero positive-degree cohomology for the full cover. This proves the auxiliary fact by induction on the number of opens. Thus horizontal cohomology of $C^{\bullet ,q}$ consists only of $\Gamma (X,{\mathcal{I}}^q)$ in degree zero. Computing the cohomology of the total complex first horizontally therefore gives $H^n(X,\mathcal{F})$, by the resolution principle for sheaf cohomology.

On the other hand, vertical cohomology issince the restricted flasque resolutions compute cohomology on each intersection. The already established affine vanishing makes all rows with $q>0$ zero. The surviving row $q=0$ is exactly the Čech cochain complex ${\check{C}}^{\bullet }(\mathcal{U},\mathcal{F})$. Computing total cohomology first vertically therefore gives ${\check{H}}^n(\mathcal{U},\mathcal{F})$. These two computations are justified by the two filtrations of the first-quadrant double complex: in every total degree only finitely many terms occur, and the cover also bounds the horizontal degree. Their edge maps give the natural identificationFor negative degrees both groups are zero by convention. In particular, degree zero is the usual identification by the sheaf gluing axiom, not merely a comparison of dimensions.

A sheaf of modules $\mathcal{F}$ over ${\mathcal{O}}_X$ is quasi-coherent when locally it admits a presentationwith arbitrary indexing sets $I,J$. On an affine chart, sheafifying the corresponding module presentation and using exactness of localization identifies its cokernel with a sheaf $\tilde{N}$. Thus one may equivalently require that locally on affine charts $\mathcal{F}$ is associated with a module. This does not impose finite generation, which belongs to the stronger coherent condition.

On the affine variety $X$ with coordinate ring $A$, put $M=\Gamma (X,\mathcal{F})$. Refine local module charts to a finite principal open cover $X={\bigcup }_{j}D(g_j)$ such that $\mathcal{F}{|}_{D(g_j)}\cong {\tilde{N}}_j$ for an $A_{g_j}$-module $N_j$. This is possible because principal opens form an open basis and $X$ is quasi-compact. On the overlap $D(g_j{g}_k)$ the sections are the corresponding further localization of $N_j$ or $N_k$. The sheaf gluing axiom gives an equalizerwhere the last map is the difference of restrictions. For any $f\in A$, localize this sequence. Exactness of localization and commutation with finite products give exactly the equalizer for the cover $D(f)={\bigcup }_{j}D(f{g}_j)$. HenceThe isomorphisms are canonical and commute with all basic-open restrictions. Since the sections of $\tilde{M}$ on these opens are $M_f$, they identify the two sheaves:Conversely $\tilde{M}$ is quasi-coherent for every $A$-module $M$, since a free presentation of $M$ gives the required sheaf presentation. Also $\Gamma (X,\tilde{M})=M$, and module homomorphisms sheafify, while a sheaf morphism is determined on every $D(f)$ by the localization of its map on global sections. This proves the affine module-sheaf equivalence.

For a short exact sequence of quasi-coherent sheaves, the sequence of their stalks is exact. Under this equivalence the stalk at $x$ is the localization of the global-section module at ${\mathfrak{m}}_x$. Exactness of module sequences can be checked at all maximal ideals, by localization detects zero elements applied to their homology, so the global-section modules also form a short exact sequence. Therefore global sections are exact on quasi-coherent sheaves over an affine variety. This is stronger than the left exactness of the global section functor on arbitrary sheaves, and follows from localization, not from assuming the cohomology vanishing still to be proved.

For an open inclusion $j:U\hookrightarrow X$, the requested sheaf isRestriction gives $\mathcal{F}\to {}_U\mathcal{F}$. This direct image from an open restriction is not extension by zero: its stalks outside $U$ may be nonzero.

The locally vanishing principle for sheaf cohomology says that a class $\xi \in H^i(X,\mathcal{F})$, $i>0$, is killed on a suitable neighbourhood of every point. Under the stated hypothesis that basis opens and their finite intersections have zero cohomology in degrees $1,\dots ,i-1$, these neighbourhoods can be chosen in that basis so that the image of $\xi$ underis zero. In particular its restriction in $H^i(U,\mathcal{F}{|}_U)$ is zero. This is a statement about individual classes; it does not assert that every locally vanishing class is already globally zero.

Here is a noncircular proof of vanishing of quasi-coherent cohomology on an affine scheme, applied to the variety. Induct on $i>0$, simultaneously for every affine variety and quasi-coherent sheaf. Assume all lower positive degrees vanish. The principal-open basis is closed under finite intersections, so it satisfies the principle's hypothesis. Given $\xi \in H^i(X,\mathcal{F})$, choose a finite principal cover $\mathcal{U}=(D(f_j))$ on which it restricts to zero.

Applying the Čech cochain complex to a flasque resolution gives the Čech lifting below the first possible local cohomology degree comparison segment, under lower-degree vanishing on all intersectionsFor completeness, the double complex has terms ${\check{C}}^p(\mathcal{U},{\mathcal{I}}^q)$. Its augmented rows are exact because each ${\mathcal{I}}^q$ is flasque, so its total cohomology is the cohomology of global sections of the resolution. The vertical cohomology on intersections in degrees $0<q<i$ is zero. Equivalently, start with local primitives of a cocycle representing $\xi$, take their differences on pairwise overlaps, and solve successively for primitives of those differences in degrees $i-1,i-2,\dots ,1$. The last difference is a Čech $i$-cocycle with values in $\mathcal{F}$. This identifies the kernel of the restriction map with the displayed Čech group. No vanishing in degree $i$ on the intersections has been assumed.

It remains a module calculation. Write $\mathcal{F}=\tilde{M}$. The augmented Čech complex isSince the $D(f_j)$ cover $X$, the $f_j$ generate the unit ideal. This complex is exact. To see it algebraically, use alternating cochains, with repeated indices giving zero. For a cocycle $c$, choose $N$ large enough to clear every $f_j$-denominator in $f_j^N{c}_{jI}$, as well as the finitely many cocycle relations; vanishing criterion in a module localization allows a further power to clear relations which initially hold only after localization. Since the ideal $(f_j^N)$ is still the unit ideal, choose $a_j$ with ${\sum }_j{a}_j{f}_j^N=1$. Defineusing those cleared representatives in $M_{f_I}$. The cocycle identity gives $\delta b=({\sum }_j{a}_j{f}_j^N)c=c$. The same argument with the augmentation gives gluing and uniqueness in degree zero. This is exactness of the unit-ideal localization Čech complex.

Thus ${\check{H}}^i(\mathcal{U},\mathcal{F})=0$, so the class $\xi$, whose restrictions were zero, is zero. The induction starts at $i=1$, when the lower-degree condition is empty. We have proved

For an affine variety, every coherent ideal sheaf is quasi-coherent; vanishing of quasi-coherent cohomology on an affine scheme therefore gives $H^1(X,\mathcal{I})=0$.

For the projective-space complement, assume $n\ge 1$ and choose two distinct closed points $P,Q\in X$. Take the ideal sheaf of two closed points $\mathcal{I}={\mathfrak{m}}_P\cap {\mathfrak{m}}_Q$ on $X$. The codimension-two extension of regular functions on a normal variety givesOne can see this directly: on every standard affine chart of ${\mathbb{P}}^n$, a rational function written in lowest terms cannot have a nonconstant denominator, because an irreducible polynomial factor of the denominator would define a pole along a codimension-one hypersurface, and such a hypersurface is not removed by $Z$. The extended function is constant because every global regular function on projective space is constant. Now the short exact sequencesends $k$ diagonally into $k^2$ on global sections. Its cokernel is $k$, and the long exact sequence in sheaf cohomology injects that cokernel into $H^1(X,\mathcal{I})$. ThusThe assumption $n\ge 1$ is necessary: ${\mathbb{P}}^0$ is already affine and has no such example.

For $U_{i_0\dots i_q}=U_{i_0}\cap \cdots \cap U_{i_q}$, the Čech cochain complex has groupsTerms in ${\delta }^2$ cancel in pairs, and Čech cohomology is $\ker \delta /\mathrm{im}\delta$. A degree-zero cocycle is exactly a family of compatible local sections. The sheaf gluing axiom gives their unique global section, proving ${\check{H}}^0(\mathcal{U},\mathcal{F})\cong \mathcal{F}(X)$. If the cover has affine finite intersections, a quasi-coherent sheaf has no higher cohomology on those intersections by vanishing of quasi-coherent cohomology on an affine scheme. The acyclic cover theorem then identifies all Čech groups with sheaf cohomology. In particular, a finite affine open cover of a separated variety has this property.

For the sheaf of units of the structure sheaf, a multiplicative degree-one cocycle is a family $g_{ij}\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$ satisfying $g_{ij}{g}_{jk}=g_{ik}$, with $g_{ji}=g_{ij}^{-1}$. It glues trivial rank-one free modules into an invertible sheaf. Changing the local frames multiplies $g_{ij}$ by a coboundary, and two sets of transition data give isomorphic line bundles precisely when their cocycles differ this way. Tensoring line bundles multiplies their cocycles. Thus line bundles trivialized by an open cover give the group isomorphismFor the remaining arguments, work over the algebraically closed ground field. On the irreducible variety $V$, put $\mathcal{Q}={\mathcal{K}}^{\ast }/{\mathcal{O}}_V^{\ast }$, a quotient of sheaves of abelian groups. A global section of $\mathcal{Q}$ is locally represented by rational functions $f_i$ whose ratios are regular units. The corresponding unit cocycle $f_j/f_i$ defines an invertible sheaf. A single global rational function has trivial cocycle. Conversely, every line bundle has a nonzero rational section: choose a nonzero vector in its one-dimensional fibre at the generic point and express it in local frames. This supplies such local $f_i$. If the associated line bundle is trivial, changing frames makes all $f_i$ restrictions of one rational function. Thereforeis exact. This is the Cartier-divisor description of the Picard group. The sheaf of nonzero rational functions on an irreducible variety is flasque, since all restrictions between nonempty open sets are the identity on $k(V{)}^{\ast }$. Apply the long exact sequence in sheaf cohomology to $1\to {\mathcal{O}}_V^{\ast }\to {\mathcal{K}}^{\ast }\to \mathcal{Q}\to 1$. Since $H^1(V,{\mathcal{K}}^{\ast })=0$, its connecting map has exactly the cokernel just computed, provingFinally the Segre description of a smooth quadric surface identifies $V$ with ${\mathbb{P}}^1\times {\mathbb{P}}^1$. The two rulings of a smooth quadric surface have classes $F_1,F_2$ generating $\mathrm{Pic}(V)\cong {\mathbb{Z}}^2$. For completeness, remove one line in each ruling: the remaining chart is the affine plane, with factorial coordinate ring $k[s,t]$ and trivial divisor class group. The localization sequence for the divisor class group makes $F_1,F_2$ generators, and their degrees on the two ruling lines prove independence. The hyperplane class, and hence the conic $C$, has bidegree $(1,1)$. By Picard-group localization on a smooth variety, the Picard group of a smooth affine quadric surface isExplicitly, the restriction of ${\mathcal{O}}_V(1,0)$ is nontrivial: if it were trivial on $U$, its rational trivialization would have divisor supported on $C$, forcing $(1,0)$ to be an integer multiple of $(1,1)$, which is impossible.

Write $U_0=D_{+}(t_0)=\mathrm{Spec}\mathbb{C}[x]$, where $x=t_1/t_0$, and $U_1=D_{+}(t_1)=\mathrm{Spec}\mathbb{C}[y]$, with $y=1/x$ on the overlap. Since $\mathcal{G}$ is a quasi-coherent sheaf, it is associated to a $\mathbb{C}[x]$-module $M$ on $U_0$. The preimages under the open inclusion $f$ of $U_0,U_1,U_0\cap U_1$ are $U_0,D(x),D(x)$, all affine schemes. Thus $f$ is an affine morphism and its direct image sheaf $f_{\ast }\mathcal{G}$ is quasi-coherent by direct image of a quasi-coherent sheaf under an affine morphism.

The two-chart cover is an acyclic cover by vanishing of quasi-coherent cohomology on an affine scheme, so the acyclic cover theorem identifies its Čech cohomology with sheaf cohomology. Its only potentially nontrivial positive-degree differential isThis is surjective because of the second summand. There are no normalized Čech terms in degree two or higher for a two-open cover, soThis is the acyclic direct image from an affine chart of the projective line. The pushforward need not itself be a flasque sheaf; the acyclic affine cover is what justifies this computation.

The resolution principle for sheaf cohomology states that an exact sequencewith $H^q(X,{\mathcal{A}}^j)=0$ for every $q>0$ computes $H^i(X,\mathcal{F})$ as the cohomology of $\Gamma (X,{\mathcal{A}}^{\bullet })$. In particular one may use a flasque resolution.

An affine morphism $\varphi :X\to Y$ is one for which the inverse image of every affine open subset of $Y$ is affine. We use vanishing of quasi-coherent cohomology on an affine scheme: a quasi-coherent sheaf has zero higher sheaf cohomology on an affine variety. Take a flasque resolution $\mathcal{F}\to {\mathcal{I}}^{\bullet }$. Each direct image sheaf ${\varphi }_{\ast }{\mathcal{I}}^j$ is flasque, because restriction on $V\subseteq U\subseteq Y$ is restriction on ${\varphi }^{-1}V\subseteq {\varphi }^{-1}U$. On an affine open $U\subseteq Y$, the augmented complex of sections isIts positive-degree cohomology vanishes by the affine theorem, and its degree-zero kernel is $\Gamma ({\varphi }^{-1}U,\mathcal{F})$. Checking exactness on this affine basis shows that ${\varphi }_{\ast }{\mathcal{I}}^{\bullet }$ is a flasque resolution of ${\varphi }_{\ast }\mathcal{F}$. The global-section complexes are identical, proving cohomology under an affine morphism:

For projective space, put $S=k[x_0,\dots ,x_n]$ with its usual grading and $U_i=\{x_i\ne 0\}$. Construct the twisting sheaf on projective space by setting its sections on $U_i$ to be the degree-$m$ part $(S_{x_i}{)}_m$, viewed as a module over $(S_{x_i}{)}_0$. It is free of rank one with generator $x_i^m$, even when $m$ is negative. On overlaps the generators differ by the unit $(x_i/x_j{)}^m$, so these modules glue to a line bundle $\mathcal{O}(m)$.

Assume first $n\ge 1$. These local modules all embed in the graded fraction field, and agreement on overlaps identifies global sections with ${\bigcap }_i(S_{x_i}{)}_m$. Since $S$ is a unique factorization domain and at least two distinct variables occur, a reduced fraction belonging to every $S_{x_i}$ has no nonconstant denominator. Thus the intersection is $S_m$. Counting monomials by stars and bars givesHere $h_n^i(m)={\dim }_k{H}^i({\mathbf{P}}^n,\mathcal{O}(m))$. The printed formula needs the convention that a counting binomial coefficient is zero when its upper integer is smaller than its nonnegative lower integer; the generalized polynomial convention for negative upper integers would give incorrect dimensions. For $n=0$, every twist on the single point ${\mathbf{P}}^0$ is trivial, so $h_0^0(m)=1$ for every $m$ and its higher cohomology is zero. This case supplies the base of dimension induction; the displayed top-cohomology formula below is for $n\ge 1$.

The standard $n+1$ affine opens have affine finite intersections. The acyclic cover theorem and affine vanishing identify sheaf cohomology with a Čech cochain complex ending in degree $n$. In particular $H^i({\mathbf{P}}^n,\mathcal{O}(m))=0$ for $i>n$.

Let $H=\{x_n=0\}\cong {\mathbf{P}}^{n-1}$, with inclusion $i$. Multiplication by $x_n$ gives the hyperplane exact sequence for twisting sheavesExactness is checked in the local trivializations, where a hyperplane equation is a non-zero-divisor and the quotient is its restriction to $H$. Since a closed immersion is an affine morphism, the first part identifies the cohomology of $i_{\ast }{\mathcal{O}}_H(m)$ with that on $H$.

For $n=1$, the long exact sequence in sheaf cohomology and $H\cong {\mathbf{P}}^0$ giveStarting from the given vanishing for sufficiently large $m$ and descending gives $h_1^1(m)=0$ for $m\ge -1$ and $h_1^1(m)=-m-1$ for $m\le -2$.